Back row: Gabe, John, Brian
Front row: emily, Sarah
Projects
emily: She has decided on a "mystery" project given to her from
László Babai.
Update: The mystery has been revealed-- emily is thinking
about the following "game." At stage 1, you have
the number 1. At stage n, you can either double the number you have
or perform any permutation of the digits. She is pondering many
questions associated with this "game" such as generalizations
to writing the numbers in other bases, figuring out the most
number of stages possible to remain within a certain length of digits,
etc.
References so far:
A Course in Combinatorics by J. H. Van Lint and R. M. Wilson
Sarah: Interested in counting how many curves can be placed
on a genus g surface that
pairwise only intersect once.
This is still an open problem in general, but there are many interesting
partial results. She has improved the quadratic lower bound found
by Farb and Leininger in 2006, as well as showing there is
an exponential upper bound on the number of such curves.
Her current resources are:
Classical Topology and Combinatorial Group Theory by John Stillwell
Riemannian Geometry by Manfredo Peridigão do Carmo
Class notes from a course taught by Benson Farb
John: Learning group theory. In particular, he proves the
Schur-Zassenhaus Theorem.
Resources:
A Course in the Theory of Groups by Derek Robinson
Groups and Representations by John Alperin and Rowen Bell.
Gabe: He will understand the ideal class group,
including the finiteness theorem as well as the unit theorem for number
fields, including computations of class numbers.
His current resources are:
Number Fields by Daniel Marcus
Class notes from a class taught by Madhav Nori
Brian: A project on the representation theory of finite groups; he
learns the basics, including Maschke's Theorem, Schur's Lemma and
character theory, including computations of some character tables.
He hopes to move on to Fulton
and Harris, possibly understanding the representation of S_n.
Current resources:
Groups and Representations by John Alperin and Rowen Bell.
Representation Theory: A First Course by William Fulton and
Joe Harris.
Quantum Game Theory (a prerequisite is some knowledge of regular old game theory--of course, you can make learning the latter part of your project)
References to topics that some mentees have expressed interest in
For representation theory:
Groups and Representations by J. L. Alperin and R. B. Bell.
To learn the basics of representation theory of finite groups, check out Sections 12 and 14-16.
Representation Theory: A First Course by W. Fulton and J. Harris.
More advanced topics than Alperin and Bell, to be read after reading the above.
Lectures 1-5 involve a slightly more high-brow study of representations of
finite groups, culminating in classifying representations of S_n and
A_n (the symmetric group on n letters and the alternating group,
respectively). To learn some Lie Theory skip ahead to Lectures 7-11, which
culminates in classifying the representations of SL_n(C)
(the group
of 2x2 matrices with determinant 1 and entries in the the complex numbers).
For number theory:
Algebraic Theory of Numbers (or Théorie Algébrique des Nombres if you want to learn some French) by P. Samuel.
An excellent book on algebraic number theory. Works through finiteness of the ideal class group and the unit theorem (with great exercises that involve many calculations) and then applications to classical problems in number theory.
Algebraic Number Theory; Proceedings of an Instructional Conference edited by J. W. S. Cassels and A. Frölich.
A much more advanced version of the above, but a real classic in algebraic number theory. It will put some hair on your chest (whether male or female).
Rational Points on Elliptic Curves by J. H. Silverman and J. Tate.
Another fantastic book with an elementary account of elliptic curves, proving most of Mordell's theorem that rational points are finiteley generated (to prove the whole thing, a prerequisite is some algebraic number theory), and then looking at elliptic curves over finite fields, which are responsible for much of modern cryptography.
The Arithmetic of Elliptic Curves by J. H. Silverman.
A more advanced version of the above, perhaps read alongside or after.
For combinatorics/graph theory:
A Course in Combinatorics by J. H. Van Lint and R. M. Wilson.
I haven't read it personally, but it's a reference to the Sodoku article listed above, and Matt Thibault highly recommends it.
